A Note on the Computation of All Zeros of Simple Quaternionic Polynomials

Janovská, Drahoslava; Opfer, Gerhard

doi:10.1137/090748871

Cited by 57 publications

(66 citation statements)

References 8 publications

(10 reference statements)

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“…26 (2016) Matrices Over Nondivision Algebras 607 are considered. More recent information on problems related to quaternions and coquaternions with extensions to other algebras can be found in papers by the current authors in [8][9][10][11][12][13].…”

Section: −I II Iii −Imentioning

confidence: 99%

Matrices Over Nondivision Algebras Without Eigenvalues

Janovská

Opfer

2015

Adv. Appl. Clifford Algebras

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Abstract. We are concerned with matrices over nondivision algebras and show by an example from an R 4 algebra that these matrices do not necessarily have eigenvalues, even if these matrices are invertible. The standard condition for eigenvectors x = 0 will be replaced by the condition that x contains at least one invertible component which is the same as x = 0 for division algebras. The topic is of principal interest, and leads to the question what qualifies a matrix over a nondivision algebra to have eigenvalues. And connected with this problem is the question, whether these matrices are diagonalizable or triangulizable and allow a Schur decomposition. There is a last section where the question whether a specific matrix A has eigenvalues is extended to all eight R 4 algebras by applying numerical means. As a curiosity we found that the considered matrix A over the algebra of tessarines, which is a commutative algebra, introduced by Cockle (Phil Mag 35(3):434-437, 1849; http:// www.oocities.org/cocklebio/), possesses eigenvalues.Mathematics Subject Classification. 11E88, 15A66, 20G20, 65F15, 65H17.Keywords. Eigenvalues of matrices over nondivision algebras, Eigenvalues of matrices over coquaternions, Split-quaternions, Eigenvalues of matrices over a general R 4 algebra, Diagonalizable and triangulizable matrices over nondivision algebras.

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Section: −I II Iii −Imentioning

confidence: 99%

Matrices Over Nondivision Algebras Without Eigenvalues

Janovská

Opfer

2015

Adv. Appl. Clifford Algebras

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“…In the most general case, incorporating arbitrary juxtapositions of the variable and coefficients, there is no fundamental theorem of algebra, since equations with no solutions can easily be constructed. Consequently, most studies consider only polynomials in which all coefficients are to the left or right of the powers of the quaternion variable, e.g., [23,26,28,32,33,36]. Even in this restricted setting, unusual features -such as the occurrence of spherical roots -may arise.…”

Section: Analysis Of the Quaternion Equationsmentioning

confidence: 99%

Tensor-product surface patches with Pythagorean-hodograph isoparametric curves

et al. 2015

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The construction of tensor-product surface patches with a family of Pythagorean-hodograph (PH) isoparametric curves is investigated. The simplest non-trivial instances, interpolating four prescribed patch boundary curves, involve degree (5, 4) tensor-product surface patches x(u, v) whose v = constant isoparametric curves are all spatial PH quintics. It is shown that the construction can be reduced to solving a novel type of quadratic quaternion equation, in which the quaternion unknown and its conjugate exhibit left and right coefficients, while the quadratic term has a coefficient interposed between them. A closedform solution for this type of equation is derived, and conditions for the existence of solutions are identified. The surfaces incorporate three residual scalar freedoms which can be exploited to improve the interior shape of the patch. The implementation of the method is illustrated through a selection of computed examples.

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“…The most important rule is rule (2.14). It allows to write col(axb) = ı 1 (a)ı 2 (b)col(x), which means that the linear mapping This was successfully applied to the solution of quaternionic, linear systems, and to finding zeros of certain quaternionic polynomials, see Janovská and Opfer, [6,7,8].…”

Section: Quaternions and Pseudoquaternions In The Matrix Space R 4×4mentioning

confidence: 99%

“…for all quaternions z ∈ H. This was used by Pogorui and Shapiro, 2004, [9] and by the present authors [7,8].…”

Section: Quaternions and Pseudoquaternions In The Matrix Space R 4×4mentioning

confidence: 99%

The Nonexistence of Pseudoquaternions in $${\mathbb{C}^{2\times 2}}$$

Janovská

Opfer

2010

Adv. Appl. Clifford Algebras

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The field of quaternions, denoted by H can be represented as an isomorphic four dimensional subspace of R 4×4 , the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in R 4×4 which is also a field and which has -in connection with the quaternions -many pleasant properties. This field is called field of pseudoquaternions. It exists in R 4×4 but not in H. It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in R 4 . And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b. Now, the field of quaternions can also be represented as an isomorphic four dimensional subspace of C 2×2 over R, the space of complex matrices with two rows and columns. We show that in this space pseudoquaternions with all the properties known from R 4×4 do not exist. However, there is a subset of C 2×2 for which some of the properties are still valid. By means of the Kronecker product we show that there is a matrix in C 4×4 which has the properties of the pseudoquaternionic matrix.Mathematics Subject Classification (2010). 11R52, 12E15.

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A Note on the Computation of All Zeros of Simple Quaternionic Polynomials

Cited by 57 publications

References 8 publications

Matrices Over Nondivision Algebras Without Eigenvalues

Matrices Over Nondivision Algebras Without Eigenvalues

Tensor-product surface patches with Pythagorean-hodograph isoparametric curves

The Nonexistence of Pseudoquaternions in $${\mathbb{C}^{2\times 2}}$$

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